# American Institute of Mathematical Sciences

July  2011, 16(1): 1-14. doi: 10.3934/dcdsb.2011.16.1

## The Euler-Maruyama approximations for the CEV model

 1 School of Mathematical Sciences, Monash University, Clayton Campus, Building 28, Wellington road, Victoria, 3800, Australia 2 School of Mathematical Sciences, Monash University, Clayton Campus, Building 28,, Wellington road, Victoria, 3800, Australia 3 Department of Engineering Systems, Tel Aviv University, Tel Aviv, Ramat Aviv, 69978, Israel

Received  April 2010 Revised  August 2010 Published  April 2011

The CEV model is given by the stochastic differential equation $X_t=X_0+\int_0^t\mu X_s ds+\int_0^t\sigma (X^+_s)^p dW_s$, $\frac{1}{2}\le p<1$. It features a non-Lipschitz diffusion coefficient and gets absorbed at zero with a positive probability. We show the weak convergence of Euler-Maruyama approximations $X_t^n$ to the process $X_t$, $0 \le t \le T$, in the Skorokhod metric, by giving a new approximation by continuous processes. We calculate ruin probabilities as an example of such approximation. The ruin probability evaluated by simulations is not guaranteed to converge to the theoretical one, because the limiting distribution is discontinuous at zero. To approximate the size of the jump at zero we use the Levy metric, and also confirm the convergence numerically.
Citation: Vyacheslav M. Abramov, Fima C. Klebaner, Robert Sh. Lipster. The Euler-Maruyama approximations for the CEV model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 1-14. doi: 10.3934/dcdsb.2011.16.1
##### References:
 [1] A. Alfonsi, Higher order discretization schemes for the CIR process: application to affine term structure and Heston models,, Mathematics of Computation, 79 (2010), 209.  doi: 10.1090/S0025-5718-09-02252-2.  Google Scholar [2] V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function,, Probability Theory and Related Fields, 104 (1996), 43.  doi: 10.1007/BF01303802.  Google Scholar [3] P. Billingsley, "Converges of Probability Measures,", John Wiley & Sons, (1968).   Google Scholar [4] M. Bossy and A. Diop, An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form $|x|^\alpha, \alpha\in [1/2, 1)$,, preprint, (2007).   Google Scholar [5] J. C. Cox, The constant elasticity of variance option. Pricing model,, The Journal of Portfolio Management, 23 (1997), 15.   Google Scholar [6] D. Dawson, http://www.math.ubc.ca/ db5d/SummerSchool09/lectures-dd/lecture4.pdf,, (Accessed on August 3, (2010).   Google Scholar [7] G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term,, Applied Stochastic Models and Data Analysis, 14 (1998), 77.  doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2.  Google Scholar [8] F. Delbaen and H. A. Shirakawa, Note of option pricing for constant elasticity of variance model,, Asia-Pacific Financial Markets, 9 (2002), 159.  doi: 10.1023/A:1024173029378.  Google Scholar [9] W. Feller, Two singular diffusion problems,, Annals of Mathematics, 54 (1951), 173.  doi: 10.2307/1969318.  Google Scholar [10] I. Gy, A note on Eulers approximations,, Potential Analysis, 8 (1998), 205.  doi: 10.1023/A:1008605221617.  Google Scholar [11] I. Gy，I. ongy and N. Krylov, Existence of strong solutions for Itó's stochastic equations via approximations,, Probability Theory and Related Fields, 105 (1996), 143.  doi: 10.1007/BF01203833.  Google Scholar [12] N. Halidias and P. Kloeden, A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient,, BIT Numerical Mathematics, 48 (2008), 51.  doi: 10.1007/s10543-008-0164-1.  Google Scholar [13] P. L. Hennequin and A. Tortrat, "Théorie des Probabilités et Quelques Applications,", Masson et Cie, (1965).   Google Scholar [14] D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process,, Computational Finance, 8 (2005), 35.   Google Scholar [15] D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations,, SIAM Journal of Numerical Analysis, 40 (2002), 1041.  doi: 10.1137/S0036142901389530.  Google Scholar [16] M. Hutzenthaler and A. Jentzen, Non-globally Lipschitz counterexamples for the stochastic Euler scheme,, preprint, ().   Google Scholar [17] A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coeffcients,, Numerische Mathematik, 112 (2009), 41.  doi: 10.1007/s00211-008-0200-8.  Google Scholar [18] Yu. Kabanov, R. Liptser and A. N. Shiryaev, Estimates of closeness in variation of probability measures (Russian),, Dokl. Akad. Nauk SSSR, 278 (1984), 265.   Google Scholar [19] F. C. Klebaner, "Introduction to Stochastic Calculus with Applications,", 2$^{nd}$ edition, (2005).   Google Scholar [20] P. E. Kloeden and K. Platten, "Numerical Solution of Stochastic Differential Equations,", Springer-Verlag, (1992).   Google Scholar [21] R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales" [Translated from the Russian by K. Dzjaparidze], Mathematics and its Applications (Soviet Series), 49 (1989).   Google Scholar [22] G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics,", Springer-Verlag, (2004).   Google Scholar [23] L. C. G. Rogers and D. Williams, "Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus,", Reprint of the 2$^{nd}$ edition, (2000).   Google Scholar [24] T. Shiga and S. Watanabe, Bessel diffusions as a one parameter family of diffusion processes,, Zeitschrift f\, 27 (1973), 37.   Google Scholar [25] D. Talay, Simulation and numerical analysis of stochastic differential systems: A review,, in, 451 (1995), 63.   Google Scholar [26] L. Yan, The Euler scheme with irregular coefficients,, Annals of Probability, 30 (2002), 1172.  doi: 10.1214/aop/1029867124.  Google Scholar [27] H. Zähle, Weak approximation of SDEs by discrete time processes,, Journal of Applied Mathematics and Stochastic Analysis, 2008 (2008).   Google Scholar

show all references

##### References:
 [1] A. Alfonsi, Higher order discretization schemes for the CIR process: application to affine term structure and Heston models,, Mathematics of Computation, 79 (2010), 209.  doi: 10.1090/S0025-5718-09-02252-2.  Google Scholar [2] V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations I. Convergence rate of the distribution function,, Probability Theory and Related Fields, 104 (1996), 43.  doi: 10.1007/BF01303802.  Google Scholar [3] P. Billingsley, "Converges of Probability Measures,", John Wiley & Sons, (1968).   Google Scholar [4] M. Bossy and A. Diop, An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form $|x|^\alpha, \alpha\in [1/2, 1)$,, preprint, (2007).   Google Scholar [5] J. C. Cox, The constant elasticity of variance option. Pricing model,, The Journal of Portfolio Management, 23 (1997), 15.   Google Scholar [6] D. Dawson, http://www.math.ubc.ca/ db5d/SummerSchool09/lectures-dd/lecture4.pdf,, (Accessed on August 3, (2010).   Google Scholar [7] G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term,, Applied Stochastic Models and Data Analysis, 14 (1998), 77.  doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2.  Google Scholar [8] F. Delbaen and H. A. Shirakawa, Note of option pricing for constant elasticity of variance model,, Asia-Pacific Financial Markets, 9 (2002), 159.  doi: 10.1023/A:1024173029378.  Google Scholar [9] W. Feller, Two singular diffusion problems,, Annals of Mathematics, 54 (1951), 173.  doi: 10.2307/1969318.  Google Scholar [10] I. Gy, A note on Eulers approximations,, Potential Analysis, 8 (1998), 205.  doi: 10.1023/A:1008605221617.  Google Scholar [11] I. Gy，I. ongy and N. Krylov, Existence of strong solutions for Itó's stochastic equations via approximations,, Probability Theory and Related Fields, 105 (1996), 143.  doi: 10.1007/BF01203833.  Google Scholar [12] N. Halidias and P. Kloeden, A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient,, BIT Numerical Mathematics, 48 (2008), 51.  doi: 10.1007/s10543-008-0164-1.  Google Scholar [13] P. L. Hennequin and A. Tortrat, "Théorie des Probabilités et Quelques Applications,", Masson et Cie, (1965).   Google Scholar [14] D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process,, Computational Finance, 8 (2005), 35.   Google Scholar [15] D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations,, SIAM Journal of Numerical Analysis, 40 (2002), 1041.  doi: 10.1137/S0036142901389530.  Google Scholar [16] M. Hutzenthaler and A. Jentzen, Non-globally Lipschitz counterexamples for the stochastic Euler scheme,, preprint, ().   Google Scholar [17] A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coeffcients,, Numerische Mathematik, 112 (2009), 41.  doi: 10.1007/s00211-008-0200-8.  Google Scholar [18] Yu. Kabanov, R. Liptser and A. N. Shiryaev, Estimates of closeness in variation of probability measures (Russian),, Dokl. Akad. Nauk SSSR, 278 (1984), 265.   Google Scholar [19] F. C. Klebaner, "Introduction to Stochastic Calculus with Applications,", 2$^{nd}$ edition, (2005).   Google Scholar [20] P. E. Kloeden and K. Platten, "Numerical Solution of Stochastic Differential Equations,", Springer-Verlag, (1992).   Google Scholar [21] R. Sh. Liptser and A. N. Shiryayev, "Theory of Martingales" [Translated from the Russian by K. Dzjaparidze], Mathematics and its Applications (Soviet Series), 49 (1989).   Google Scholar [22] G. N. Milstein and M. V. Tretyakov, "Stochastic Numerics for Mathematical Physics,", Springer-Verlag, (2004).   Google Scholar [23] L. C. G. Rogers and D. Williams, "Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus,", Reprint of the 2$^{nd}$ edition, (2000).   Google Scholar [24] T. Shiga and S. Watanabe, Bessel diffusions as a one parameter family of diffusion processes,, Zeitschrift f\, 27 (1973), 37.   Google Scholar [25] D. Talay, Simulation and numerical analysis of stochastic differential systems: A review,, in, 451 (1995), 63.   Google Scholar [26] L. Yan, The Euler scheme with irregular coefficients,, Annals of Probability, 30 (2002), 1172.  doi: 10.1214/aop/1029867124.  Google Scholar [27] H. Zähle, Weak approximation of SDEs by discrete time processes,, Journal of Applied Mathematics and Stochastic Analysis, 2008 (2008).   Google Scholar
 [1] Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366 [2] Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331 [3] Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325 [4] Xiuli Xu, Xueke Pu. Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 987-1010. doi: 10.3934/dcdsb.2020150 [5] Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 [6] Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021019 [7] Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345 [8] Gheorghe Craciun, Jiaxin Jin, Casian Pantea, Adrian Tudorascu. Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1305-1335. doi: 10.3934/dcdsb.2020164 [9] Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077 [10] Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020356 [11] Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 [12] Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 [13] Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020400 [14] Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004 [15] Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316 [16] Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160 [17] Zhihua Liu, Yayun Wu, Xiangming Zhang. Existence of periodic wave trains for an age-structured model with diffusion. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021009 [18] Hui Zhao, Zhengrong Liu, Yiren Chen. Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021011 [19] Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021017 [20] Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

2019 Impact Factor: 1.27

## Metrics

• HTML views (0)
• Cited by (4)

• on AIMS